THC Science

The Fibonacci sequence is constant-recursive: each element of the sequence is the sum of the previous two.

Hasse diagram of some subsets of constant-recursive sequences, ordered by inclusion

In mathematics and theoretical computer science, a constant-recursive sequence or linear recurrence sequence is an infinite sequence of numbers in which each number in the sequence is equal to a linear combination of one or more of its immediate predecessors.

The most famous example of a constant-recursive sequence is the Fibonacci sequence, in which each number is the sum of the previous two; formally, we say that it satisfies the linear recurrence relation $F(n)=F(n-1)+F(n-2)$ , where $F(n)$ is the $n$ th Fibonacci number. Generalizing this idea, a constant-recursive sequence $s(n)$ satisfies a formula of the form

s(n)=c_{1}s(n-1)+c_{2}s(n-2)+\dots +c_{d}s(n-d),

where $c_{i}$ are constants. In addition to the Fibonacci sequence, constant-recursive sequences include many other well-known sequences, such as arithmetic progressions, geometric progressions, and polynomials.

Constant-recursive sequences arise in combinatorics and the theory of finite differences; in algebraic number theory, due to the relation of the sequence to the roots of a polynomial; in the analysis of algorithms in the running time of simple recursive functions; and in formal language theory, where they count strings up to a given length in a regular language. Constant-recursive sequences are closed under important mathematical operations such as pointwise addition, pointwise multiplication, and Cauchy product. A seminal result is the Skolem–Mahler–Lech theorem, which states that the zeros of a constant-recursive sequence have a regularly repeating (eventually periodic) form. On the other hand, the Skolem problem, which asks for algorithm to determine whether a linear recurrence has at least one zero, remains one of the unsolved problems in mathematics.

Definition

An order-d homogeneous linear recurrence relation is an equation of the form

s(n)=c_{1}s(n-1)+c_{2}s(n-2)+\dots +c_{d}s(n-d),

where the d coefficients $c_{1},c_{2},\dots ,c_{d}$ are coefficients ranging over the integers, rational numbers, algebraic numbers, real numbers, or complex numbers.

A sequence $s(0),s(1),s(2),\dots$ (ranging over the same domain as the coefficients) is a constant-recursive sequence if there is an order-d homogeneous linear recurrence with constant coefficients that it satisfies for all $n\geq d$ . Note that this definition allows eventually-periodic sequences such as $1,0,0,0,\ldots$ and $0,1,0,0,\ldots$ , which are disallowed by some authors.^[1] These sequences can be excluded by requiring that $c_{d}\neq 0$ .

The order of a constant-recursive sequence is the smallest $d\geq 1$ such that the sequence satisfies an order-d homogeneous linear recurrence, or $d=0$ for the everywhere-zero sequence.

Equivalent definitions

In terms of vector spaces

A sequence $s(n)_{n\geq 0}$ is constant-recursive if and only if the set of sequences

\{s(n+r)_{n\geq 0}:r\geq 0\}

is contained in a vector space whose dimension is finite, i.e. a finite-dimensional subspace of $\mathbb {C} ^{\mathbb {N} }$ closed under the left-shift operator. This is because the order- $d$ linear recurrence relation can be understood as a proof of linear dependence between the vectors $s(n+r)_{n\geq 0}$ for $r=0,\ldots ,d$ . An extension of this argument shows that the order of the sequence is equal to the dimension of the vector space generated by $s(n+r)_{n\geq 0}$ for all $r$ .

In terms of non-homogeneous linear recurrences

A non-homogeneous linear recurrence is an equation of the form

s(n)=c_{1}s(n-1)+c_{2}s(n-2)+\dots +c_{d}s(n-d)+c

where $c$ is an additional constant. Any sequence satisfying a non-homogeneous linear recurrence is constant-recursive. This is because subtracting the equation for $s(n-1)$ from the equation for $s(n)$ yields a homogeneous recurrence for $s(n)-s(n-1)$ , from which we can solve for $s(n)$ to obtain

s(n)=(c_{1}+1)s(n-1)+(c_{2}-c_{1})s(n-2)+\dots +(c_{d}-c_{d-1})s(n-d)-c_{d}s(n-d-1).

Examples

Fibonacci sequence

The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... of Fibonacci numbers is constant-recursive of order 2 because it satisfies the recurrence $F(n)=F(n-1)+F(n-2)$ with $F(0)=0,F(1)=1$ . For example, $F(2)=F(1)+F(0)=1+0=1$ and $F(6)=F(5)+F(4)=5+3=8$ .

Lucas sequences

The sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... (sequence A000032 in the OEIS) of Lucas numbers satisfies the same recurrence as the Fibonacci sequence but with initial conditions $L(0)=2$ and $L(1)+1$ . More generally, every Lucas sequence is constant-recursive of order 2.

Arithmetic progressions

For any $a$ and any $r\neq 0$ , the arithmetic progression $a,a+r,a+2r,\ldots$ is constant-recursive of order 2, because it satisfies $s(n)=2s(n-1)-s(n-2)$ . Generalizing this, see polynomial sequences below.

Geometric progressions

For any $a\neq 0$ and $r$ , the geometric progression $a,ar,ar^{2},\ldots$ is constant-recursive of order 1, because it satisfies $s(n)=rs(n-1)$ . This includes, for example, the sequence 1, 2, 4, 8, 16, ... as well as the rational number sequence $1,{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{8}},{\frac {1}{16}},...$ .

Eventually periodic sequences

A sequence that is eventually periodic with period length $\ell$ is constant-recursive, since it satisfies $s(n)=s(n-\ell )$ for all $n\geq d$ , where the order $d$ is the length of the initial segment including the first repeating block. Examples of such sequences are 1, 0, 0, 0, ... (order 1) and 1, 6, 6, 6, ... (order 2).

Polynomial sequences

For any polynomial $s(n)$ , the sequence of its values is a constant-recursive sequence. If the degree of the polynomial is $d$ , the sequence satisfies a recurrence of order $d+1$ , with coefficients given by the corresponding element of the binomial transform.^[2] The first few such equations are

s(n)=1\cdot s(n-1)

for a degree 0 (that is, constant) polynomial,

s(n)=2\cdot s(n-1)-1\cdot s(n-2)

for a degree 1 or less polynomial,

s(n)=3\cdot s(n-1)-3\cdot s(n-2)+1\cdot s(n-3)

for a degree 2 or less polynomial, and

s(n)=4\cdot s(n-1)-6\cdot s(n-2)+4\cdot s(n-3)-1\cdot s(n-4)

for a degree 3 or less polynomial.

A sequence obeying the order-d equation also obeys all higher order equations. These identities may be proved in a number of ways, including via the theory of finite differences.^[3] Any sequence of $d+1$ integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order $d+1$ . If the initial conditions lie on a polynomial of degree $d-1$ or less, then the constant-recursive sequence also obeys a lower order equation.

Enumeration of words in a regular language

Let $L$ be a regular language, and let $s(n)$ be the number of words of length $n$ in $L$ . Then $s(n)_{n\geq 0}$ is constant-recursive. For example, $s(n)=2^{n}$ for the language of all binary strings, $s(n)=1$ for the language of all unary strings, and $s(n)=F(n+2)$ for the language of all binary strings that do not have two consecutive ones. More generally, any function accepted by a weighted automaton over the unary alphabet $\Sigma =\{a\}$ over the semiring $(\mathbb {R} ,+,\times )$ is constant-recursive.

Other examples

The sequences of Jacobsthal numbers, Padovan numbers, Pell numbers, and Perrin numbers are constant-recursive.

Closed-form characterization

Constant-recursive sequences admit the following unique closed form characterization using exponential polynomials: every constant-recursive sequence can be written in the form

s(n)=z(n)+k_{1}(n)r_{1}^{n}+k_{2}(n)r_{2}^{n}+\cdots +k_{e}(n)r_{e}^{n},

where

$z(n)$ is a sequence which is zero for all but finitely many $n$ ;
$k_{1}(n),k_{2}(n),\ldots ,k_{e}(n)$ are complex polynomials; and
$r_{1},r_{2},\ldots ,r_{k}$ are distinct complex number constants.

This characterization is exact: every sequence that can be written in the above form is constant-recursive.

For example, the Fibonacci number $F(n)$ is written in this form using Binet's formula:

F(n)={\frac {1}{\sqrt {5}}}\varphi ^{n}+{\frac {1}{\sqrt {5}}}\psi ^{n},

where $\varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\ldots$ is the Golden ratio and $\psi ={\frac {-1}{\varphi }}$ , both roots of the equation $x^{2}-x-1=0$ . In this case, $e=2$ , $z(n)=0$ for all $n$ , $k_{1}(n)=k_{2}(n)={\frac {1}{\sqrt {5}}}$ (constant polynomials), $r_{1}=\varphi$ , and $r_{2}=\psi$ . Notice that though the original sequence was over the integers, the closed form solution involves real or complex roots. In general, for sequences of integers or rationals, the closed formula will use algebraic numbers.

The complex numbers $r_{1},\ldots ,r_{n}$ are derived as the roots of the characteristic polynomial (or "auxiliary polynomial") of the recurrence:

x^{d}-c_{1}x^{d-1}-\dots -c_{d-1}x-c_{d}

whose coefficients are the same as those of the recurrence. If the $d$ roots $r_{1},r_{2},\dots ,r_{d}$ are all distinct, then the polynomials $k_{i}(n)$ are all constants, which can be determined from the initial values of the sequence. The term $z(n)$ is only needed when $c(d)\neq 0$ ; if $c(d)=0$ then it corrects for the fact that some initial values may be exceptions to the general recurrence. In particular, $z(n)=0$ for all $n\geq d$ , the order of the sequence.

In the general case where the roots of the characteristic polynomial are not necessarily distinct, and $r_{i}$ is a root of multiplicity $m$ , the term $r_{i}^{n}$ is multiplied by a degree- $(m-1)$ polynomial in $n$ (i.e. $k_{i}(n)$ in the formula has degree $m$ ). For instance, if the characteristic polynomial factors as $(x-r)^{3}$ , with the same root r occurring three times, then the $n$ th term is of the form

s(n)=(a+bn+cn^{2})r^{n}.

^[4]

Generating function view

A sequence is constant-recursive precisely when its generating function

\sum _{n\geq 0}s(n)x^{n}=s(0)+s(1)x^{1}+s(2)x^{2}+s(3)x^{3}+\cdots

is a rational function. The denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence.^[5] For example, the generating function of the Fibonacci sequence is

{\frac {x}{1-x-x^{2}}}.

In general, multiplying a generating function by the polynomial

1-c_{1}x^{1}-c_{2}x^{2}-\dots -c_{d}x^{d}

yields a series

\left(s(0)+s(1)x^{1}+s(2)x^{2}+\cdots \right)\left(1-c_{1}x^{1}-c_{2}x^{2}-\dots -c_{d}x^{d}\right)=\left(b_{0}+b_{1}x^{1}+b_{2}x^{2}+\cdots \right),

where

b_{n}=s(n)-c_{1}s(n-1)-c_{2}s(n-2)-\dots -c_{d}s(n-d).

If $s(n)$ satisfies the recurrence relation

s(n)=c_{1}s(n-1)+c_{2}s(n-2)+\dots +c_{d}s(n-d),

then $b_{n}=0$ for all $n\geq d$ . In other words,

\left(s(0)+s(1)x^{1}+s(2)x^{2}+\cdots \right)\left(1-c_{1}x^{1}-c_{2}x^{2}-\dots -c_{d}x^{d}\right)=\left(b_{0}+b_{1}x^{1}+b_{2}x^{2}+\dots +b_{d-1}x^{d-1}\right),

so we obtain the rational function

\sum _{n\geq 0}s(n)x^{n}={\frac {b_{0}+b_{1}x^{1}+b_{2}x^{2}+\dots +b_{d-1}x^{d-1}}{1-c_{1}x^{1}-c_{2}x^{2}-\dots -c_{d}x^{d}}}.

In the special case of a periodic sequence satisfying $s(n)=s(n-d)$ for $n\geq d$ , the generating function is

{\begin{aligned}{\frac {s(0)+s(1)x^{1}+\dots +s(d-1)x^{d-1}}{1-x^{d}}}=&\left(s(0)+s(1)x^{1}+\dots +s(d-1)x^{d-1}\right)+\left(s(0)+s(1)x^{1}+\dots +s(d-1)x^{d-1}\right)x^{d}+{}\\&\left(s(0)+s(1)x^{1}+\dots +s(d-1)x^{d-1}\right)x^{2d}+\cdots \end{aligned}}

by expanding the geometric series.

The generating function of the Catalan numbers is not a rational function, which implies that the Catalan numbers do not satisfy a linear recurrence with constant coefficients.

Derivation of the closed form from the generating function

The closed form can be derived from the generating function via partial fraction decomposition. Specifically, if the generating function is written as

{\frac {f(x)}{g(x)}}=p(x)+\sum _{i}{\frac {f_{i}(x)}{g_{i}(x)}}

then the polynomial $p(x)$ gives the initial set of corrections $z(n)$ , the denominator $g_{i}(x)=(x-r_{i})^{m}$ gives rise to the exponential term $r_{i}^{n}$ , and the numerator of smaller degree $f_{i}(x)$ gives rise to the polynomial coefficient $k_{i}(n)$ .

Closure properties

The termwise addition or multiplication of two constant-recursive sequences is again constant-recursive. This follows from the characterization in terms of exponential polynomials.

The Cauchy product of two constant-recursive sequences is constant-recursive. This follows from the characterization in terms of rational generating functions.

Generalizations

A natural generalization is obtained by relaxing the condition that the coefficients of the recurrence are constants. If the coefficients are allowed to be polynomials, then one obtains holonomic sequences.

A $k$ -regular sequence satisfies linear recurrences with constant coefficients, but the recurrences take a different form. Rather than $s(n)$ being a linear combination of $s(m)$ for some integers $m$ that are close to $n$ , each term $s(n)$ in a $k$ -regular sequence is a linear combination of $s(m)$ for some integers $m$ whose base- $k$ representations are close to that of $n$ . Constant-recursive sequences can be thought of as $1$ -regular sequences, where the base-1 representation of $n$ consists of $n$ copies of the digit $1$ .

Notes

^ Kauers, Manuel; Paule, Peter (2010-12-01). The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates. Springer Vienna. ISBN 978-3-7091-0444-6.
^ Boyadzhiev, Boyad (2012). "Close Encounters with the Stirling Numbers of the Second Kind" (PDF). Math. Mag. 85: 252–266.
^ Jordan, Charles; Jordán, Károly (1965). Calculus of Finite Differences. American Mathematical Soc. pp. 9–11. ISBN 978-0-8284-0033-6. See formula on p.9, top.
^ Greene, Daniel H.; Knuth, Donald E. (1982), "2.1.1 Constant coefficients – A) Homogeneous equations", Mathematics for the Analysis of Algorithms (2nd ed.), Birkhäuser, p. 17.
^ Martino, Ivan; Martino, Luca (2013-11-14). "On the variety of linear recurrences and numerical semigroups". Semigroup Forum. 88 (3): 569–574. arXiv:1207.0111. doi:10.1007/s00233-013-9551-2. ISSN 0037-1912.

References

Brousseau, Alfred (1971). Linear Recursion and Fibonacci Sequences. Fibonacci Association.
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics: A Foundation for Computer Science (2 ed.). Addison-Wesley. ISBN 978-0-201-55802-9.

External links

"OEIS Index Rec". OEIS index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)

[1] Kauers, Manuel; Paule, Peter (2010-12-01). The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates. Springer Vienna. ISBN 978-3-7091-0444-6.

[2] Boyadzhiev, Boyad (2012). "Close Encounters with the Stirling Numbers of the Second Kind" (PDF). Math. Mag. 85: 252–266.

[3] Jordan, Charles; Jordán, Károly (1965). Calculus of Finite Differences. American Mathematical Soc. pp. 9–11. ISBN 978-0-8284-0033-6. See formula on p.9, top.

[4] Greene, Daniel H.; Knuth, Donald E. (1982), "2.1.1 Constant coefficients – A) Homogeneous equations", Mathematics for the Analysis of Algorithms (2nd ed.), Birkhäuser, p. 17.

[5] Martino, Ivan; Martino, Luca (2013-11-14). "On the variety of linear recurrences and numerical semigroups". Semigroup Forum. 88 (3): 569–574. arXiv:1207.0111. doi:10.1007/s00233-013-9551-2. ISSN 0037-1912.

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 :<math>s(n) = (a + b n + c n^2) r^n.</math><ref>{{citation|contribution=2.1.1 Constant coefficients – A) Homogeneous equations|title=Mathematics for the Analysis of Algorithms|first1=Daniel H.|last1=Greene|first2=Donald E.|last2=Knuth|author2-link=Donald Knuth| edition=2nd| publisher=Birkhäuser |page=17 | year=1982}}.</ref>
-== Generating function characterization ==
+=== Generating function view ===
 A sequence is constant-recursive precisely when its [[generating function]]